{
  "id": "2005.10931",
  "version": "v1",
  "published": "2020-05-21T22:41:07.000Z",
  "updated": "2020-05-21T22:41:07.000Z",
  "title": "On linear sets of minimum size",
  "authors": [
    "Dibyayoti Jena",
    "Geertrui Van de Voorde"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "An $\\mathbb{F}_q$-linear set of rank $k$ on a projective line $\\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J. Comb. Theory, Ser: A 164 (2019), 109-124.]). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between $k/2$ and $k-1$. Our construction extends the known examples of linear sets of size $q^{k-1}+1$ in $\\mathrm{PG}(1,q^h)$ constructed for $k=h=4$ [G. Bonoli and O. Polverino, $\\mathbb{F}_q$-Linear blocking sets in $\\mathrm{PG}(2,q^4)$, Innov. Incidence Geom. 2 (2005), 35--56.] and $k=h$ in [G. Lunardon and O. Polverino. Blocking sets of size $q^t+q^{t-1}+1$. J. Comb. Theory, Ser: A 90 (2000), 148-158.]. We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small $k$, we investigate whether all linear sets of size $q^{k-1}+1$ arise from our construction. Finally, we modify our construction to define linear sets of size $q^{k-1}+q^{k-2}+\\ldots+q^{k-l}+1$ in $\\mathrm{PG}(l,q)$. This leads to new infinite families of small minimal blocking sets which are not of R\\'edei type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-21T22:41:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E20"
    ],
    "keywords": [
      "minimum size",
      "define linear sets",
      "small minimal blocking sets",
      "heaviest point",
      "construction extends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}